Yeah, sorry, it's tough to be clear without pictures !
The wire to the battery light on the dash is the exciter wire... current passes thru the dash light and excites the alternator field windings.
So, with no wire from the alternator to the dash light, alternator will likely not start the charging cycle, and as you've said you won't have a warning light when the belt breaks etc.
If your alternator only has one thin wire coming out of it then that's the exciter lead that should go to the dash light. Can't really count on the colors matching if you think the PO was messing with things.
If you had two thin wires coming out of the alt one would be the exciter and one the W, but it doesn't sound like this is the case.
Bottom line... the thick wires from the alt go to the battery connection at the starter, and if you only have one thin wire from the alt it is most likely the exciter wire and needs to go to the light on the dash. I say "most likely" since it is possible that the PO cut off the exciter and left the W wire.
The exciter wire read 12V btw because that's the voltage that excites the alternator.
Clear as mud ??!!
One last thought.... if the above is still unclear (highly possible !) then I really suggest posting a couple of pictures... we can photoshop 'em with arrows to show what goes where.